Where \(F(T)(\equiv K(0,T;0,0))\)  denotes the fluctuation integral of the free particle confined to the plane. In the following one solves the integral Eq. (REF ) by expanding the fluctuation \(\delta \gamma (t)\)  in a complex Fourier series. This is indeed possible since \(\delta \gamma (t)\)  describes a loop in the complex plane. It is therefore possible to continue \(\delta \gamma (t)\)  to a T-periodic (complex-valued) function on \(\mathbb {R}\) . The author borrowed the idea of Fourier expanding a function describing a closed path in the plane from Hurwitz' elegant solution of the isoperimetric problem.[1]}(See also Ref.[2]})
